Is "al-Hakim" a name or a title, and is what my friend mentioned about the meaning of the title and the condition for "al-Hujjah" being memorizing thirty thousand hadiths correct?

The author of Al-Mustadrak is named Muhammad ibn Abdullah ibn Muhammad ibn Hamduwayh ibn Nasih ibn Al-Hakam, and Al-Hakim is his title. Scholarly titles such as Al-Hafiz, Al-Hujjah, and Al-Hakim are used by scholars for those who meet their specific conditions. Al-Hafiz is someone whose knowledge encompasses one hundred thousand hadiths, both text and chain of narration. Al-Hujjah is someone whose knowledge encompasses three hundred thousand hadiths in the same way. Al-Hakim is someone whose knowledge encompasses all narrated hadiths. It is also said that Al-Hakim, the author of Al-Mustadrak, was given this title because he served as a judge.